The Generalized Terwilliger Algebra of the Hypercube

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra $\mathcal T$ of a distance-regular graph $\Gamma$. For any vertex $x$ of $\Gamma,$ there is a surjective algebra homomorphism $\natural$ from $\mathcal T$ to the Terwilliger algebra $T(x)$. If $\Gamma$ is complete, then $\natural$ is an isomorphism. If $\Gamma$ is not complete, then $\natural$ may or may not be an isomorphism, and in general the details are unknown. We show that if $\Gamma$ is a hypercube, then the algebra homomorphism $\natural:\mathcal T \to T(x)$ is an isomorphism for all vertices $x$ of $\Gamma$.Comment: 24 page

Hypercubes, Leonard triples and the anticommutator spin algebra

This paper is about three classes of objects: Leonard triples, distance-regular graphs and the modules for the anticommutator spin algebra. Let \K denote an algebraically closed field of characteristic zero. Let $V$ denote a vector space over \K with finite positive dimension. A Leonard triple on $V$ is an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis for $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. The Leonard triples of interest to us are said to be totally B/AB and of Bannai/Ito type. Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the anticommutator spin algebra $\mathcal{A}$, the unital associative \K-algebra defined by generators $x,y,z$ and relations$$xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.$$ Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter $D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp. $\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp. $\tilde{Q}_{D}$). We obtain the following. When $D$ is even (resp. odd), we show that there exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp. $\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency matrices respectively. We classify the resulting irreducible $\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show that actions of the adjacency, dual adjacency and weighted adjacency matrices for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp. $\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.Comment: arXiv admin note: text overlap with arXiv:0705.0518 by other author

Entanglement of Free Fermions on Johnson Graphs

Free fermions on Johnson graphs $J(n,k)$ are considered and the entanglement entropy of sets of neighborhoods is computed. For a subsystem composed of a single neighborhood, an analytical expression is provided by the decomposition in irreducible submodules of the Terwilliger algebra of $J(n,k)$ embedded in two copies of $\mathfrak{su}(2)$. For a subsytem composed of multiple neighborhoods, the construction of a block-tridiagonal operator which commutes with the entanglement Hamiltonian is presented, its usefulness in computing the entropy is stressed and the area law pre-factor is discussed.Comment: 24 page

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page